Theorem r19.23 3314

Description: Restricted quantifier version of 19.23 2211. See r19.23v 3279 for a version requiring fewer axioms. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 8-Oct-2016.)

Hypothesis

Ref	Expression
r19.23.1	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
r19.23	⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))

Proof of Theorem r19.23

Step	Hyp	Ref	Expression
1		r19.23.1	. 2 ⊢ Ⅎ𝑥𝜓
2		r19.23t 3313	. 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 208  Ⅎwnf 1784  ∀wral 3138  ∃wrex 3139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785  df-ral 3143  df-rex 3144

This theorem is referenced by:  rexlimi  3315  ss2iundf  40024  iunssf  41372